WORST_CASE(Omega(n^1),O(n^2))
proof of input_2IGvvllHW8.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 356 ms]
    (18) BOUNDS(1, n^2)
(19) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
    (21) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRelTRS
    (23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) typed CpxTrs
    (27) OrderProof [LOWER BOUND(ID), 13 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 292 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 120 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   minus(x, y) -> cond(gt(x, y), x, y)
   cond(false, x, y) -> 0
   cond(true, x, y) -> s(minus(x, s(y)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(0, z0) -> c3
   GT(s(z0), 0) -> c4
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(0, z0) -> c3
   GT(s(z0), 0) -> c4
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, cond_3, gt_2

Defined Pair Symbols:   MINUS_2, COND_3, GT_2

Compound Symbols:   c_2, c1, c2_1, c3, c4, c5_1


----------------------------------------

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   COND(false, z0, z1) -> c1
   GT(s(z0), 0) -> c4
   GT(0, z0) -> c3

----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, cond_3, gt_2

Defined Pair Symbols:   MINUS_2, COND_3, GT_2

Compound Symbols:   c_2, c2_1, c5_1


----------------------------------------

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))

----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   gt_2

Defined Pair Symbols:   MINUS_2, COND_3, GT_2

Compound Symbols:   c_2, c2_1, c5_1


----------------------------------------

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))

The (relative) TRS S consists of the following rules:

   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(9) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1)) [1]
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1))) [1]
   GT(s(z0), s(z1)) -> c5(GT(z0, z1)) [1]
   gt(0, z0) -> false [0]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(z0, z1) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1)) [1]
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1))) [1]
   GT(s(z0), s(z1)) -> c5(GT(z0, z1)) [1]
   gt(0, z0) -> false [0]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(z0, z1) [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c
c :: c2 -> c5 -> c
COND :: true:false -> s:0 -> s:0 -> c2
gt :: s:0 -> s:0 -> true:false
GT :: s:0 -> s:0 -> c5
true :: true:false
c2 :: c -> c2
s :: s:0 -> s:0
c5 :: c5 -> c5
0 :: s:0
false :: true:false

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   gt(v0, v1) -> null_gt [0]
   COND(v0, v1, v2) -> null_COND [0]
   GT(v0, v1) -> null_GT [0]

And the following fresh constants:    null_gt, null_COND, null_GT, const

----------------------------------------

(14)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1)) [1]
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1))) [1]
   GT(s(z0), s(z1)) -> c5(GT(z0, z1)) [1]
   gt(0, z0) -> false [0]
   gt(s(z0), 0) -> true [0]
   gt(s(z0), s(z1)) -> gt(z0, z1) [0]
   gt(v0, v1) -> null_gt [0]
   COND(v0, v1, v2) -> null_COND [0]
   GT(v0, v1) -> null_GT [0]

The TRS has the following type information:
MINUS :: s:0 -> s:0 -> c
c :: c2:null_COND -> c5:null_GT -> c
COND :: true:false:null_gt -> s:0 -> s:0 -> c2:null_COND
gt :: s:0 -> s:0 -> true:false:null_gt
GT :: s:0 -> s:0 -> c5:null_GT
true :: true:false:null_gt
c2 :: c -> c2:null_COND
s :: s:0 -> s:0
c5 :: c5:null_GT -> c5:null_GT
0 :: s:0
false :: true:false:null_gt
null_gt :: true:false:null_gt
null_COND :: c2:null_COND
null_GT :: c5:null_GT
const :: c

Rewrite Strategy: INNERMOST
----------------------------------------

(15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 2
   0 => 0
   false => 1
   null_gt => 0
   null_COND => 0
   null_GT => 0
   const => 0

----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   COND(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   COND(z, z', z'') -{ 1 }-> 1 + MINUS(z0, 1 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   GT(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GT(z, z') -{ 1 }-> 1 + GT(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MINUS(z, z') -{ 1 }-> 1 + COND(gt(z0, z1), z0, z1) + GT(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   gt(z, z') -{ 0 }-> gt(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   gt(z, z') -{ 0 }-> 2 :|: z = 1 + z0, z0 >= 0, z' = 0
   gt(z, z') -{ 0 }-> 1 :|: z0 >= 0, z = 0, z' = z0
   gt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(17) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V4),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[fun1(V1, V, V4, Out)],[V1 >= 0,V >= 0,V4 >= 0]).
eq(start(V1, V, V4),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V4),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[gt(V3, V2, Ret010),fun1(Ret010, V3, V2, Ret01),fun2(V3, V2, Ret1)],[Out = 1 + Ret01 + Ret1,V1 = V3,V2 >= 0,V = V2,V3 >= 0]).
eq(fun1(V1, V, V4, Out),1,[fun(V6, 1 + V5, Ret11)],[Out = 1 + Ret11,V1 = 2,V5 >= 0,V6 >= 0,V = V6,V4 = V5]).
eq(fun2(V1, V, Out),1,[fun2(V8, V7, Ret12)],[Out = 1 + Ret12,V7 >= 0,V1 = 1 + V8,V8 >= 0,V = 1 + V7]).
eq(gt(V1, V, Out),0,[],[Out = 1,V9 >= 0,V1 = 0,V = V9]).
eq(gt(V1, V, Out),0,[],[Out = 2,V1 = 1 + V10,V10 >= 0,V = 0]).
eq(gt(V1, V, Out),0,[gt(V11, V12, Ret)],[Out = Ret,V12 >= 0,V1 = 1 + V11,V11 >= 0,V = 1 + V12]).
eq(gt(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(fun1(V1, V, V4, Out),0,[],[Out = 0,V16 >= 0,V4 = V17,V15 >= 0,V1 = V16,V = V15,V17 >= 0]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V4,Out),[V1,V,V4],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [fun2/3]
1. recursive  : [gt/3]
2. recursive [non_tail] : [fun/3,fun1/4]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun2/3
1. SCC is partially evaluated into gt/3
2. SCC is partially evaluated into fun/3
3. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations fun2/3 
* CE 9 is refined into CE [14] 
* CE 8 is refined into CE [15] 


### Cost equations --> "Loop" of fun2/3 
* CEs [15] --> Loop 10 
* CEs [14] --> Loop 11 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [10]: [V,V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V
    V1


### Specialization of cost equations gt/3 
* CE 13 is refined into CE [16] 
* CE 11 is refined into CE [17] 
* CE 10 is refined into CE [18] 
* CE 12 is refined into CE [19] 


### Cost equations --> "Loop" of gt/3 
* CEs [19] --> Loop 12 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [18] --> Loop 15 

### Ranking functions of CR gt(V1,V,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR gt(V1,V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations fun/3 
* CE 7 is refined into CE [20,21,22] 
* CE 6 is refined into CE [23,24,25,26,27,28,29,30] 


### Cost equations --> "Loop" of fun/3 
* CEs [30] --> Loop 16 
* CEs [26,28] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [23,25,27,29] --> Loop 19 
* CEs [21] --> Loop 20 
* CEs [22] --> Loop 21 
* CEs [20] --> Loop 22 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [20,21]: [V1-V]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [20,21]:
  - RF of loop [20:1,21:1]:
    V1-V


### Specialization of cost equations start/3 
* CE 1 is refined into CE [31] 
* CE 2 is refined into CE [32,33,34] 
* CE 3 is refined into CE [35,36,37,38,39,40] 
* CE 4 is refined into CE [41,42] 
* CE 5 is refined into CE [43,44,45,46,47] 


### Cost equations --> "Loop" of start/3 
* CEs [36,37,38,44] --> Loop 23 
* CEs [32,33,34] --> Loop 24 
* CEs [31,35,39,40,41,42,43,45,46,47] --> Loop 25 

### Ranking functions of CR start(V1,V,V4) 

#### Partial ranking functions of CR start(V1,V,V4) 


Computing Bounds
=====================================

#### Cost of chains of fun2(V1,V,Out):
* Chain [[10],11]: 1*it(10)+0
  Such that:it(10) =< Out

  with precondition: [Out>=1,V1>=Out,V>=Out] 

* Chain [11]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of gt(V1,V,Out):
* Chain [[12],15]: 0
  with precondition: [Out=1,V1>=1,V>=V1] 

* Chain [[12],14]: 0
  with precondition: [Out=2,V>=1,V1>=V+1] 

* Chain [[12],13]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [15]: 0
  with precondition: [V1=0,Out=1,V>=0] 

* Chain [14]: 0
  with precondition: [V=0,Out=2,V1>=1] 

* Chain [13]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun(V1,V,Out):
* Chain [[20,21],19]: 4*it(20)+1*s(3)+1
  Such that:aux(1) =< V1
aux(4) =< V1-V
it(20) =< aux(4)
s(3) =< it(20)*aux(1)

  with precondition: [V>=1,Out>=3,V1>=V+1] 

* Chain [[20,21],17]: 4*it(20)+1*s(3)+2*s(4)+1
  Such that:aux(5) =< V1
aux(6) =< V1-V
s(4) =< aux(5)
it(20) =< aux(6)
s(3) =< it(20)*aux(5)

  with precondition: [V>=1,Out>=4,V1>=V+1] 

* Chain [[20,21],16]: 4*it(20)+1*s(3)+1*s(6)+1
  Such that:aux(7) =< V1
aux(8) =< V1-V
s(6) =< aux(7)
it(20) =< aux(8)
s(3) =< it(20)*aux(7)

  with precondition: [V>=1,Out>=4,V1>=V+2] 

* Chain [22,[20,21],19]: 4*it(20)+1*s(3)+3
  Such that:aux(9) =< V1
it(20) =< aux(9)
s(3) =< it(20)*aux(9)

  with precondition: [V=0,V1>=2,Out>=5] 

* Chain [22,[20,21],17]: 6*it(20)+1*s(3)+3
  Such that:aux(10) =< V1
it(20) =< aux(10)
s(3) =< it(20)*aux(10)

  with precondition: [V=0,V1>=2,Out>=6] 

* Chain [22,[20,21],16]: 5*it(20)+1*s(3)+3
  Such that:aux(11) =< V1
it(20) =< aux(11)
s(3) =< it(20)*aux(11)

  with precondition: [V=0,V1>=3,Out>=6] 

* Chain [22,19]: 3
  with precondition: [V=0,Out=3,V1>=1] 

* Chain [22,17]: 1*s(4)+1*s(5)+3
  Such that:s(4) =< 1
s(5) =< V1

  with precondition: [V=0,Out=4,V1>=1] 

* Chain [22,16]: 1*s(6)+3
  Such that:s(6) =< 1

  with precondition: [V=0,Out=4,V1>=2] 

* Chain [19]: 1
  with precondition: [Out=1,V1>=0,V>=0] 

* Chain [18]: 1
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [17]: 1*s(4)+1*s(5)+1
  Such that:s(5) =< V1
s(4) =< V

  with precondition: [Out>=2,V1+1>=Out,V+1>=Out] 

* Chain [16]: 1*s(6)+1
  Such that:s(6) =< V

  with precondition: [Out>=2,V1>=V+1,V+1>=Out] 


#### Cost of chains of start(V1,V,V4):
* Chain [25]: 4*s(38)+12*s(39)+3*s(40)+3*s(43)+1
  Such that:s(37) =< V1-V
aux(17) =< V1
aux(18) =< V
s(38) =< aux(17)
s(43) =< aux(18)
s(39) =< s(37)
s(40) =< s(39)*aux(17)

  with precondition: [V1>=0,V>=0] 

* Chain [24]: 4*s(47)+12*s(48)+3*s(49)+2*s(52)+2
  Such that:s(46) =< V-V4
s(51) =< V4+1
aux(19) =< V
s(47) =< aux(19)
s(48) =< s(46)
s(49) =< s(48)*aux(19)
s(52) =< s(51)

  with precondition: [V1=2,V>=0,V4>=0] 

* Chain [23]: 16*s(53)+2*s(55)+3*s(58)+3
  Such that:s(54) =< 1
aux(20) =< V1
s(53) =< aux(20)
s(55) =< s(54)
s(58) =< s(53)*aux(20)

  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V,V4): 
-------------------------------------
* Chain [25] with precondition: [V1>=0,V>=0] 
    - Upper bound: 4*V1+1+3*V1*nat(V1-V)+3*V+nat(V1-V)*12 
    - Complexity: n^2 
* Chain [24] with precondition: [V1=2,V>=0,V4>=0] 
    - Upper bound: 4*V+2+3*V*nat(V-V4)+(2*V4+2)+nat(V-V4)*12 
    - Complexity: n^2 
* Chain [23] with precondition: [V=0,V1>=1] 
    - Upper bound: 16*V1+5+3*V1*V1 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V4): max([4*V1+max([12*V1+4+3*V1*V1,3*V1*nat(V1-V)+3*V+nat(V1-V)*12]),4*V+1+3*V*nat(V-V4)+nat(V4+1)*2+nat(V-V4)*12])+1 
Asymptotic class: n^2 
* Total analysis performed in 310 ms.


----------------------------------------

(18)
BOUNDS(1, n^2)

----------------------------------------

(19) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(0, z0) -> c3
   GT(s(z0), 0) -> c4
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))
S tuples:
   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(0, z0) -> c3
   GT(s(z0), 0) -> c4
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, cond_3, gt_2

Defined Pair Symbols:   MINUS_2, COND_3, GT_2

Compound Symbols:   c_2, c1, c2_1, c3, c4, c5_1


----------------------------------------

(21) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(22)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(0, z0) -> c3
   GT(s(z0), 0) -> c4
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))

The (relative) TRS S consists of the following rules:

   minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
   cond(false, z0, z1) -> 0
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(23) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(24)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
   COND(false, z0, z1) -> c1
   COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
   GT(0', z0) -> c3
   GT(s(z0), 0') -> c4
   GT(s(z0), s(z1)) -> c5(GT(z0, z1))

The (relative) TRS S consists of the following rules:

   minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
   cond(false, z0, z1) -> 0'
   cond(true, z0, z1) -> s(minus(z0, s(z1)))
   gt(0', z0) -> false
   gt(s(z0), 0') -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(25) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(26)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GT(0', z0) -> c3
GT(s(z0), 0') -> c4
GT(s(z0), s(z1)) -> c5(GT(z0, z1))
minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> false:true
GT :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
s :: s:0' -> s:0'
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5

----------------------------------------

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, gt, GT, minus

They will be analysed ascendingly in the following order:
gt < MINUS
GT < MINUS
gt < minus

----------------------------------------

(28)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GT(0', z0) -> c3
GT(s(z0), 0') -> c4
GT(s(z0), s(z1)) -> c5(GT(z0, z1))
minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> false:true
GT :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
s :: s:0' -> s:0'
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5


Generator Equations:
gen_s:0'6_6(0) <=> 0'
gen_s:0'6_6(+(x, 1)) <=> s(gen_s:0'6_6(x))
gen_c3:c4:c57_6(0) <=> c3
gen_c3:c4:c57_6(+(x, 1)) <=> c5(gen_c3:c4:c57_6(x))


The following defined symbols remain to be analysed:
gt, MINUS, GT, minus

They will be analysed ascendingly in the following order:
gt < MINUS
GT < MINUS
gt < minus

----------------------------------------

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) -> false, rt in Omega(0)

Induction Base:
gt(gen_s:0'6_6(0), gen_s:0'6_6(0)) ->_R^Omega(0)
false

Induction Step:
gt(gen_s:0'6_6(+(n9_6, 1)), gen_s:0'6_6(+(n9_6, 1))) ->_R^Omega(0)
gt(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) ->_IH
false

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(30)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GT(0', z0) -> c3
GT(s(z0), 0') -> c4
GT(s(z0), s(z1)) -> c5(GT(z0, z1))
minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> false:true
GT :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
s :: s:0' -> s:0'
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5


Lemmas:
gt(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) -> false, rt in Omega(0)


Generator Equations:
gen_s:0'6_6(0) <=> 0'
gen_s:0'6_6(+(x, 1)) <=> s(gen_s:0'6_6(x))
gen_c3:c4:c57_6(0) <=> c3
gen_c3:c4:c57_6(+(x, 1)) <=> c5(gen_c3:c4:c57_6(x))


The following defined symbols remain to be analysed:
GT, MINUS, minus

They will be analysed ascendingly in the following order:
GT < MINUS

----------------------------------------

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GT(gen_s:0'6_6(n290_6), gen_s:0'6_6(n290_6)) -> gen_c3:c4:c57_6(n290_6), rt in Omega(1 + n290_6)

Induction Base:
GT(gen_s:0'6_6(0), gen_s:0'6_6(0)) ->_R^Omega(1)
c3

Induction Step:
GT(gen_s:0'6_6(+(n290_6, 1)), gen_s:0'6_6(+(n290_6, 1))) ->_R^Omega(1)
c5(GT(gen_s:0'6_6(n290_6), gen_s:0'6_6(n290_6))) ->_IH
c5(gen_c3:c4:c57_6(c291_6))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(32)
Complex Obligation (BEST)

----------------------------------------

(33)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GT(0', z0) -> c3
GT(s(z0), 0') -> c4
GT(s(z0), s(z1)) -> c5(GT(z0, z1))
minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> false:true
GT :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
s :: s:0' -> s:0'
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5


Lemmas:
gt(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) -> false, rt in Omega(0)


Generator Equations:
gen_s:0'6_6(0) <=> 0'
gen_s:0'6_6(+(x, 1)) <=> s(gen_s:0'6_6(x))
gen_c3:c4:c57_6(0) <=> c3
gen_c3:c4:c57_6(+(x, 1)) <=> c5(gen_c3:c4:c57_6(x))


The following defined symbols remain to be analysed:
GT, MINUS, minus

They will be analysed ascendingly in the following order:
GT < MINUS

----------------------------------------

(34) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(35)
BOUNDS(n^1, INF)

----------------------------------------

(36)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, z1) -> c(COND(gt(z0, z1), z0, z1), GT(z0, z1))
COND(false, z0, z1) -> c1
COND(true, z0, z1) -> c2(MINUS(z0, s(z1)))
GT(0', z0) -> c3
GT(s(z0), 0') -> c4
GT(s(z0), s(z1)) -> c5(GT(z0, z1))
minus(z0, z1) -> cond(gt(z0, z1), z0, z1)
cond(false, z0, z1) -> 0'
cond(true, z0, z1) -> s(minus(z0, s(z1)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
MINUS :: s:0' -> s:0' -> c
c :: c1:c2 -> c3:c4:c5 -> c
COND :: false:true -> s:0' -> s:0' -> c1:c2
gt :: s:0' -> s:0' -> false:true
GT :: s:0' -> s:0' -> c3:c4:c5
false :: false:true
c1 :: c1:c2
true :: false:true
c2 :: c -> c1:c2
s :: s:0' -> s:0'
0' :: s:0'
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
minus :: s:0' -> s:0' -> s:0'
cond :: false:true -> s:0' -> s:0' -> s:0'
hole_c1_6 :: c
hole_s:0'2_6 :: s:0'
hole_c1:c23_6 :: c1:c2
hole_c3:c4:c54_6 :: c3:c4:c5
hole_false:true5_6 :: false:true
gen_s:0'6_6 :: Nat -> s:0'
gen_c3:c4:c57_6 :: Nat -> c3:c4:c5


Lemmas:
gt(gen_s:0'6_6(n9_6), gen_s:0'6_6(n9_6)) -> false, rt in Omega(0)
GT(gen_s:0'6_6(n290_6), gen_s:0'6_6(n290_6)) -> gen_c3:c4:c57_6(n290_6), rt in Omega(1 + n290_6)


Generator Equations:
gen_s:0'6_6(0) <=> 0'
gen_s:0'6_6(+(x, 1)) <=> s(gen_s:0'6_6(x))
gen_c3:c4:c57_6(0) <=> c3
gen_c3:c4:c57_6(+(x, 1)) <=> c5(gen_c3:c4:c57_6(x))


The following defined symbols remain to be analysed:
MINUS, minus